Mathematical aspects of numerical solution of hyperbolic systems

A number of physical phenomena are described by nonlinear hyperbolic equations. Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems. Construction of such methods for systems more complicated that the Euler gas dynamic systems requires the investigation of existence and uniqueness of self-similar solution to be used in the development of discontinuity-capturing high-resolution numerical methods. This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion. We discuss these problems in the application to the mahnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic waves in magnetics. Paper presented to the 7th International Conference on Hyperbolic Problems. To be published by Birkhauser.